One-Variable
Equations & Inequalities
Master creating and solving equations and inequalities from linear, quadratic, rational, and exponential functions in one variable.
Algebra 2
What Is a One-Variable Equation?
A one-variable equation is a mathematical statement that two expressions are equal, where only one unknown quantity (variable) appears. Your job: find every value of that variable that makes both sides equal.
Equation
Uses \( = \) to state two expressions have the same value.
Variable
A letter (often \( x \)) representing an unknown number we solve for.
Solution
The value(s) of the variable that make the equation true.
What Is a One-Variable Inequality?
An inequality compares two expressions using symbols other than \( = \). Instead of one exact answer, the solution is typically a range (or set) of values.
\( < \) and \( > \)
Strict inequalities: the variable is less than or greater than a value (open circle on a number line).
\( \leq \) and \( \geq \)
Non-strict: the variable can also equal the boundary value (closed circle on a number line).
The Four Function Families
The standard requires you to create equations and inequalities from four types of functions. Each family has a distinct shape and solving strategy.
Linear
\( ax + b = c \)
Straight line, one solution.
Straight line, one solution.
Quadratic
\( ax^2 + bx + c = 0 \)
Parabola, up to two solutions.
Parabola, up to two solutions.
Rational
\( \dfrac{a}{x} = b \)
Hyperbola, check for excluded values.
Hyperbola, check for excluded values.
Exponential
\( a \cdot b^x = c \)
Rapid growth/decay, use logarithms.
Rapid growth/decay, use logarithms.
Visualizing Each Family
Creating Equations from Word Problems
The most important skill is translating a real-world scenario into a mathematical equation or inequality. Follow a systematic approach every time.
1
Identify the unknown. Assign a variable (e.g., \( x \)) to the quantity you need to find.
2
Translate phrases to math. "is" → \( = \), "at most" → \( \leq \), "sum of" → \( + \), "product of" → \( \times \).
3
Write the equation or inequality. Combine the expressions using the correct relation symbol.
4
Solve and verify. Solve algebraically, then check the answer in the original context.
Translation Cheat Sheet
| English Phrase | Math Symbol | Example |
|---|---|---|
| "is", "equals", "the result is" | \( = \) | \( x + 3 = 10 \) |
| "is less than" | \( < \) | \( 2x < 8 \) |
| "is at most", "no more than" | \( \leq \) | \( x \leq 5 \) |
| "is greater than", "exceeds" | \( > \) | \( x > 12 \) |
| "is at least", "no less than" | \( \geq \) | \( x \geq 100 \) |
| "twice", "double" | \( 2x \) | \( 2x + 1 = 9 \) |
| "the square of" | \( x^2 \) | \( x^2 = 49 \) |
Equations vs Inequalities
| Feature | Equation | Inequality |
|---|---|---|
| Symbol | \( = \) | \( <,\; >,\; \leq,\; \geq \) |
| Solution type | Specific value(s) | Range / interval |
| Graph on number line | Point(s) | Ray or segment |
| Multiply by negative | No sign change | Flip the inequality |
| Notation | \( x = 3 \) | \( x \in (-\infty, 3) \) |
Solving Strategy: Step by Step
1
Simplify each side. Distribute, combine like terms, clear fractions by multiplying by the LCD.
2
Isolate the variable term. Use addition/subtraction to move constants to one side.
3
Solve for the variable. Divide (or factor for quadratics). For inequalities, remember to flip if dividing by a negative.
4
Check domain restrictions. Rational equations: exclude values making the denominator zero. Exponential: check \( b > 0, b \neq 1 \).
5
Verify. Substitute back into the original equation/inequality.
Example 1: Linear Equation
1
Define variable: Let \( x = \) number of texts.
2
Write equation: \( 25 + 0.10x = 43 \)
3
Solve: \( 0.10x = 18 \) → \( x = 180 \)
4
Verify: \( 25 + 0.10(180) = 25 + 18 = 43 \) ✓
Example 2: Quadratic Equation
1
Set up: \( -16t^2 + 64t + 5 = 69 \)
2
Rearrange: \( -16t^2 + 64t - 64 = 0 \) → divide by \( -16 \): \( t^2 - 4t + 4 = 0 \)
3
Factor: \( (t - 2)^2 = 0 \) → \( t = 2 \)
4
Verify: \( -16(2^2) + 64(2) + 5 = -64 + 128 + 5 = 69 \) ✓
Example 3: Rational Equation
1
Domain restrictions: \( x \neq 1 \) and \( x \neq -2 \).
2
Cross-multiply: \( 3(x+2) = 6(x-1) \).
3
Expand: \( 3x + 6 = 6x - 6 \) → \( 12 = 3x \) → \( x = 4 \).
4
Verify: \( \frac{3}{4-1} = 1 \) and \( \frac{6}{4+2} = 1 \). Both sides equal 1 ✓. And \( 4 \neq 1, -2 \) ✓.
Example 4: Exponential Equation
1
Model: \( P(t) = 200 \cdot 2^{t/5} \). Set up inequality: \( 200 \cdot 2^{t/5} > 3200 \).
2
Divide: \( 2^{t/5} > 16 \). Since \( 16 = 2^4 \): \( 2^{t/5} > 2^4 \).
3
Solve exponents: \( \frac{t}{5} > 4 \) → \( t > 20 \).
4
Interpret: The population exceeds 3200 after 20 years.
Example 5: Quadratic Inequality
1
Factor: \( (x-2)(x-3) \leq 0 \). Roots: \( x = 2 \) and \( x = 3 \).
2
Sign analysis: The parabola opens upward (\( a = 1 > 0 \)), so it is below or on the \( x \)-axis between the roots.
3
Solution: \( 2 \leq x \leq 3 \), or in interval notation: \( [2, 3] \).
Common Mistakes to Avoid
✕ Forgetting to Flip
When multiplying or dividing an inequality by a negative, the sign must reverse. \( -2x > 6 \) becomes \( x < -3 \), not \( x > -3 \).
✕ Ignoring Domain
Rational equations can produce extraneous solutions. Always check that your answer does not make a denominator zero.
✕ Dropping Solutions
Quadratic equations can have two solutions. Don't stop after finding one root: factor completely or use the quadratic formula.
Summary: Solving by Type
| Type | Method | Watch For |
|---|---|---|
| Linear | Isolate \( x \) with inverse operations | Flip sign if dividing by a negative |
| Quadratic | Factor, complete the square, or quadratic formula | Two roots; sign chart for inequalities |
| Rational | Cross-multiply or multiply by LCD | Excluded values; extraneous solutions |
| Exponential | Rewrite with same base or use logarithms | \( \text{Base} > 1 \) vs \( 0 < \text{base} < 1 \) changes inequality direction |
Question 1 of 5
A gym charges a \( \$50 \) registration fee plus \( \$30 \) per month. You have budgeted at most \( \$290 \). Which inequality models the number of months \( m \) you can afford?
"At most \$290" means the total cost must be less than or equal to 290. The total cost is \( 50 + 30m \), giving \( 50 + 30m \leq 290 \). Solving: \( m \leq 8 \).
Question 2 of 5
What are the solutions to \( x^2 - 7x + 10 = 0 \)?
Factor: \( (x-2)(x-5) = 0 \). We need two numbers that multiply to \( +10 \) and add to \( -7 \): that's \( -2 \) and \( -5 \). Setting each factor to zero: \( x = 2 \) or \( x = 5 \).
Question 3 of 5
Solve: \( \dfrac{5}{x} = \dfrac{10}{x+3} \). What is \( x \)?
Cross-multiply: \( 5(x+3) = 10x \) → \( 5x + 15 = 10x \) → \( 15 = 5x \) → \( x = 3 \). Check domain: \( x \neq 0 \) and \( x \neq -3 \). Since \( 3 \) is valid, the answer is \( x = 3 \).
Question 4 of 5
If \( -3x + 9 \geq 0 \), what is the solution?
Subtract 9: \( -3x \geq -9 \). Divide by \( -3 \) and flip the inequality: \( x \leq 3 \). Remember: dividing by a negative reverses the sign.
Question 5 of 5
An investment of \( \$1000 \) grows by \( 5\% \) each year. Which equation models the amount \( A \) after \( t \) years?
A \( 5\% \) annual growth means each year the amount is multiplied by \( 1 + 0.05 = 1.05 \). The exponential model is \( A = 1000 \cdot 1.05^t \). Choice (a) is linear, (b) uses the wrong base, and (c) models decay.